Highly composite number



A highly composite number is a positive integer that has more divisors than any smaller positive integer. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are.

Ramanujan wrote a paper on highly composite numbers in 1915.

The mathematician Jean-Pierre Kahane suggested that Plato must have known about highly composite numbers as he deliberately chose such a number, 5040 (= 7!), as the ideal number of citizens in a city. Furthermore, Vardoulakis and Pugh's paper delves into a similar inquiry concerning the number 5040.

Examples
The first 41 highly composite numbers are listed in the table below. The number of divisors is given in the column labeled d(n). Asterisks indicate superior highly composite numbers.

The divisors of the first 19 highly composite numbers are shown below.

The table below shows all 72 divisors of 10080 by writing it as a product of two numbers in 36 different ways.

The 15,000th highly composite number can be found on Achim Flammenkamp's website. It is the product of 230 primes:


 * $$a_0^{14} a_1^9 a_2^6 a_3^4 a_4^4 a_5^3 a_6^3 a_7^3 a_8^2 a_9^2 a_{10}^2 a_{11}^2 a_{12}^2 a_{13}^2 a_{14}^2 a_{15}^2 a_{16}^2 a_{17}^2 a_{18}^{2} a_{19} a_{20} a_{21}\cdots a_{229},$$

where $$a_n$$ is the $$n$$th successive prime number, and all omitted terms (a22 to a228) are factors with exponent equal to one (i.e. the number is $$2^{14} \times 3^{9}  \times 5^6  \times \cdots \times 1451$$). More concisely, it is the product of seven distinct primorials:


 * $$b_0^5 b_1^3 b_2^2 b_4 b_7 b_{18} b_{229},$$

where $$b_n$$ is the primorial $$a_0a_1\cdots a_n$$.

Prime factorization
Roughly speaking, for a number to be highly composite it has to have prime factors as small as possible, but not too many of the same. By the fundamental theorem of arithmetic, every positive integer n has a unique prime factorization:


 * $$n = p_1^{c_1} \times p_2^{c_2} \times \cdots \times p_k^{c_k}$$

where $$p_1 < p_2 < \cdots < p_k$$ are prime, and the exponents $$c_i$$ are positive integers.

Any factor of n must have the same or lesser multiplicity in each prime:


 * $$p_1^{d_1} \times p_2^{d_2} \times \cdots \times p_k^{d_k}, 0 \leq d_i \leq c_i, 0 < i \leq k$$

So the number of divisors of n is:


 * $$d(n) = (c_1 + 1) \times (c_2 + 1) \times \cdots \times (c_k + 1).$$

Hence, for a highly composite number n,


 * the k given prime numbers pi must be precisely the first k prime numbers (2, 3, 5, ...); if not, we could replace one of the given primes by a smaller prime, and thus obtain a smaller number than n with the same number of divisors (for instance 10 = 2 &times; 5 may be replaced with 6 = 2 &times; 3; both have four divisors);
 * the sequence of exponents must be non-increasing, that is $$c_1 \geq c_2 \geq \cdots \geq c_k$$; otherwise, by exchanging two exponents we would again get a smaller number than n with the same number of divisors (for instance 18 = 21 × 32 may be replaced with 12 = 22 × 31; both have six divisors).

Also, except in two special cases n = 4 and n = 36, the last exponent ck must equal 1. It means that 1, 4, and 36 are the only square highly composite numbers. Saying that the sequence of exponents is non-increasing is equivalent to saying that a highly composite number is a product of primorials or, alternatively, the smallest number for its prime signature.

Note that although the above described conditions are necessary, they are not sufficient for a number to be highly composite. For example, 96 = 25 × 3 satisfies the above conditions and has 12 divisors but is not highly composite since there is a smaller number (60) which has the same number of divisors.

Asymptotic growth and density
If Q(x) denotes the number of highly composite numbers less than or equal to x, then there are two constants a and b, both greater than 1, such that
 * $$(\log x)^a \le Q(x) \le (\log x)^b \, .$$

The first part of the inequality was proved by Paul Erdős in 1944 and the second part by Jean-Louis Nicolas in 1988. We have


 * $$1.13862 < \liminf \frac{\log Q(x)}{\log\log x} \le 1.44 \ $$

and


 * $$\limsup \frac{\log Q(x)}{\log\log x} \le 1.71 \ .$$

Related sequences
Highly composite numbers greater than 6 are also abundant numbers. One need only look at the three largest proper divisors of a particular highly composite number to ascertain this fact. It is false that all highly composite numbers are also Harshad numbers in base 10. The first highly composite number that is not a Harshad number is 245,044,800; it has a digit sum of 27, which does not divide evenly into 245,044,800.

10 of the first 38 highly composite numbers are superior highly composite numbers. The sequence of highly composite numbers is a subset of the sequence of smallest numbers k with exactly n divisors.

Highly composite numbers whose number of divisors is also a highly composite number are
 * 1, 2, 6, 12, 60, 360, 1260, 2520, 5040, 55440, 277200, 720720, 3603600, 61261200, 2205403200, 293318625600, 6746328388800, 195643523275200.

It is extremely likely that this sequence is complete.

A positive integer n is a largely composite number if d(n) ≥ d(m) for all m ≤ n. The counting function QL(x) of largely composite numbers satisfies
 * $$(\log x)^c \le \log Q_L(x) \le (\log x)^d \ $$

for positive c and d with $$0.2 \le c \le d \le 0.5$$.

Because the prime factorization of a highly composite number uses all of the first k primes, every highly composite number must be a practical number. Due to their ease of use in calculations involving fractions, many of these numbers are used in traditional systems of measurement and engineering designs.